Who says they trust him? Politics is not about the trust.
Ferelden was neck deep in trouble. There was a civil war and a horde of darkspawn roaming the country. Loghain didn't to much to stop the Blight, and some bannors were already destroyed.
That would have very bad influence on business and the economics of the country. And guess who controlled that?
The nobles needed the Blight to end as soon as possible. Eamon offered a solution.
Also, I don't believe they heard about it only at the Landsmeet. 'Preparing the landsmeet' doesn't include only purely organisational and administrative tasks, like invitation or accomodation, that's just marginal. No, 'preparing the landsmeet' - and Eamon was busy with it for quite some time - means political negotiations with possible allies.
So on the landsmeet, Eamon already had a backup, plus Alistair had the Warden's support - and at that moment the Warden represented the biggest military force in the country. To stand up against Alistair on that landsmeet would be too dangerous.
Later, when the war is over, the situation more-less stable, the Warden's army out of the way - then there's time and opportunity to start trying to decrease Eamon's and Alistair's power, deglorify them, point out the mistakes they made, their faults, etc.